We report on combined measurements of heat and charge transport through a single-electron transistor. The device acts as a heat switch actuated by the voltage applied on the gate. The Wiedemann-Franz law for the ratio of heat and charge conductances is found to be systematically violated away from the charge degeneracy points. The observed deviation agrees well with the theoretical expectation. With large temperature drop between the source and drain, the heat current away from degeneracy deviates from the standard quadratic dependence in the two temperatures. The flow of heat at the microscopic level is a fundamentally important issue, in particular if it can be converted into free energy via thermoelectric effects [1]. The ability of most conductors to sustain heat flow is linked to the electrical conductance σ via the Wiedemann-Franz law: κ/σ = L 0 T , where κ is the heat conductance,3e 2 the Lorenz number and T the temperature. While the understanding of quantum charge transport in nano-electronic devices has reached a great level of maturity, heat transport experiments are lagging far behind [2], for two essential reasons: (i) unlike charge, heat is not conserved and (ii) there is no simple thermal equivalent to the ammeter. Heat transport can nevertheless give insight to phenomena that charge transport is blind to [3,4] and, remarkably, a series of experiments has demonstrated the very universality of the quantization of heat conductance, regardless of the carriers statistics [3][4][5][6][7][8][9][10][11].As device dimensions are reduced, electron interactions gain capital importance, leading to Coulomb blockade in mesoscopic devices in which a small island is connected by tunnel junctions. A metallic island connected to a source and a drain through tunnel junctions exceeding the Klitzing resistance R K = h/e 2 and under the influence of a gate electric field constitutes a Single-Electron Transistor (SET) [12]. The charging energy of the island by a single electron writes E C = e 2 /2C where C is the total capacitance of the island. It defines the temperature and bias thresholds below which single-electron physics appears. In the regime where charge transport is governed by unscreened Coulomb interactions, the question of the associated heat flow has been addressed by several theoretical studies [13][14][15][16][17][18][19][20]. The Wiedemann-Franz law is expected to hold in an SET only at the charge degener- acy points in the limit of small transparency, where the effective transport channel is free from interactions, and is violated otherwise. In this Letter, we report on the measurements of both the heat and charge conduction through a metallic SET, with both quantities displaying a marked gate modulation. A strong deviation from the Wiedemann-Franz law is observed when the transport through the SET is arXiv:1704.02622v1 [cond-mat.mes-hall]

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